It is not difficult to show that this agrees with Kibble's suggestion from [1]}. For reference, the non-linear equation put forth by Kibble in a flat space-time takes the form
\(i\partial _{t}\Psi _{t} = \int d^{3}x\left(\hat{H}^{0}_{x}+V_{x}^{NL}\right)\Psi _{t}\) 
